Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals

Abstract

The Barbero-Immirzi parameter $\gamma$ appears in the \emph{real} connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a one-parameter quantization ambiguity in Loop Quantum Gravity. In this paper we investigate the conditions under which $\gamma$ will have physical effects in Euclidean Quantum Gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with non-zero cosmological constant on a manifold with boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition, and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that $\gamma$ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well $\gamma$ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black-hole mergers.

18 pages; v2: references added; to appear in Classical and Quantum Gravity; v3: typos corrected; minor revisions to match published version

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• Dark Energy from Topology
• Comments On Torsion and MacDowell-Mansouri gravity
• A tale of two theories of gravity in asymptotically Anti-de Sitter spacetime
• Manifold invariants affect dynamics in ADS gravity
• A topological origin for Dark Energy